Chapter 9.2, Problem 47E

To determine

The corrections to be made in hypothesis test run to check the claim that more than three quarters of school engaged in bullying behavior.

Explanation of Solution

Given:A hypothesis test is run to verify the claim that more than three quarters of school engaged in bullying behavior.

Sample size = n = 558

Number of students who behaved with physical aggression in last 30 days = 445

Concept used:The conditions to satisfy to run one sample Z test

1) The sample should be random

2) np = 10

3) n(1-p) = 10

4) There should be at least 10% of people of population in the sample.

Test statistic

  $Z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

If p value is less than the level of significance, we have sufficient evidence to reject the null hypothesis.

Calculation:Let p be the proportion of students in middle school who engaged in bullying

Null Hypothesis and alternate hypothesis are shown below

  $\begin{array}{l}{H}_0:p=0.75\\ H_a:p>0.75\end{array}$

The proportion of students in middle school who engaged in bullying

from sample is

  $\hat{p}=\frac{445}{558}=0.797$

  $p_0=0.75\left(fromnullhypothesis\right)$

The sample of 558 students in middle school are selected randomly

  $\begin{array}{l}n{p}_0=558*0.75=418\\ n{p}_0\ge 10\\ n(1-p_0)=558*(1-0.75)=140\\ n(1-p_0)\ge 10\end{array}$

Here there will be at least 5580 students in middle school, so 10% condition is satisfied.

So here we run a one sample Z test. Test statistic is calculated as shown below

  $\begin{array}{l}Z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\\ Z=\frac{0.797-0.75}{\sqrt{\frac{0.75(1-0.75)}{558}}}=\frac{0.047}{0.018}=2.6\end{array}$

The p value is P(Z = 2.6)

using standard Z table we get

  $\begin{array}{l}P\left(Z<2.6\right)=0.9953\\ P\left(Z\ge 2.6\right)=1–P\left(Z<2.6\right)\\ =1–0.9953=0.0047\end{array}$

The probability that null hypothesis is true is 0.0047

The p value(0.0047) is less than level of significance 0.05, So we have sufficient evidence to reject the null hypothesis.

Conclusion:

At 5% level of significance, we have sufficient evidence that more than three quarters of school students engaged in bullying behavior.